Empirical Optimization of the Source-Surface Height in the PFSS extrapolation

TL;DR

This work addresses how to set the PFSS source-surface height $R_{ m SS}$ by matching PFSS open flux to in-situ open flux using a long-term data set (2006–2023) and ADAPT-GONG magnetograms. It finds that $R_{ m SS}^{\rm opt}$ is not simply tied to solar activity and is strongly influenced by the input magnetogram, with distinct relationships: near solar maximum $R_{ m SS}^{\rm opt}$ correlates with the mean coronal field $\left< B_{r,\rm cor} \right>$, while near solar minimum it tracks dipolar dominance $f_{ m dip}$. The authors derive a two-branch empirical formula, combining both indicators via a maximum operation, to reproduce the long-term trend with typical yearly-scale accuracy ($\sim\pm 0.3R_\odot$). This approach provides a practical path for magnetogram-driven PFSS extrapolations in space-weather forecasting and highlights the need to consider both surface-field strength and global topology in determining $R_{ m SS}$.

Abstract

The potential field source surface (PFSS) method is a widely used magnetic field extrapolation technique in the space weather community. The only free parameter in the PFSS method is the source-surface height ($R_{\rm SS}$), beyond which all field lines are open. Although $R_{\rm SS}$ is known to vary with solar activity, there is no consensus on how to determine it for a given surface magnetic field distribution. In this study, we investigate the nature of $R_{\rm SS}$ using a long-period (2006-2023) data, covering two solar minima and one maximum. We adopt ADAPT-GONG magnetograms and determine $R_{\rm SS}$ by matching the open flux estimated from observations at 1 au with that calculated using the PFSS method. Our analysis reveals that $R_{\rm SS}$ increases slightly after the solar minima and around the solar maximum, and that it can be characterized by both the mean unsigned photospheric magnetic field strength and the dipolarity parameter $f_{\rm dip}$, defined as $f_{\rm dip} = B_{\rm dip}^2/(B_{\rm dip}^2 + B_{\rm quad}^2 + B_{\rm oct}^2)$, with $B_{\rm dip}$, $B_{\rm quad}$, and $B_{\rm oct}$ denoting the magnitudes of dipolar, quadrupolar, and octupolar components of photospheric radial magnetic field, respectively. Our results suggest that $R_{\rm SS}$ does not exhibit a simple monotonic dependence on the solar activity and must be determined by properly considering both surface magnetic field strength and global field structure.

Figures (9)

• Figure 1: Process of calculating the optimal value of the source-surface height ($R_{\rm SS}^{\rm opt}$). The top panel presents the time evolution of the 3-rotation averaged open fluxes estimated from in-situ observations (gray) and the PFSS extrapolations using ADAPT-GONG as input, with different source-surface radii (blue: $R_{\rm SS}/R_\odot = 1.5$, orange: $R_{\rm SS}/R_\odot = 1.9$, green: $R_{\rm SS}/R_\odot = 2.3$). The bottom panel presents the temporal evolution of the optimal source-surface height (ADAPT-GONG) computed using Equation \ref{['eq:rssopt_interpolation']}.
• Figure 2: Long-term evolution of solar magnetic activity and optimal source-surface height derived from various magnetograms. Top: monthly-averaged (gray) and 13-month smoothed (black) sunspot numbers. Middle and bottom: optimal source-surface heights estimated from observed (middle) and ADAPT (bottom) magnetograms. Red, purple, green, and blue lines denote KPVT, SOHO/MDI, GONG, and SDO/HMI, respectively. Transparent diamonds show values for each Carrington Rotation; solid lines indicate 13-CR averages.
• Figure 3: Relation between optimal source-surface height ($R_{\rm SS}^{\rm opt}$) and mean unsigned coronal radial field strength ($\left< B_{r, {\rm cor}} \right>$). Data from 2007 to 2023 are shown in the upper panel, and data from 2011 to 2017 in the lower panel. Each panel displays the value of the Peason correlation coefficient (PCC).
• Figure 4: Time evolution of the multipolar components of the photospheric magnetic field and their relation to $R_{\rm SS}$. Top: Time evolution of dipolar (red, defined in Equation \ref{['eq:dipolar_field_definition']}), quadrupolar (green, Equation \ref{['eq:quadrupolar_field_definition']}), and octupolar (blue, Eq. (3)) field strengths. Bottom: Comparison of the time evolution of dipolar field dominance $f_{\mathrm{dip}}$ (red, Eq. (4)) and optimal source-surface height $R_{\mathrm{ss}}^{\mathrm{opt}}$ (black). In both panels, transparent diamonds show values for each Carrington rotation; solid lines represent the 13-CR running average.
• Figure 5: Scatter plots between the optimal source-surface height and characteristic parameters. The top two panels show $R_\mathrm{ss}^\mathrm{opt}$ versus the mean unsigned radial field strength at the photosphere, while the bottom panels show $R_\mathrm{ss}^\mathrm{opt}$ versus the dominance of the dipole component. The left column covers the full period (2007–2023), and the right column shows results for solar maximum (2012–2016) and minimum (2017–2021) phases.